By Alexander Kechris, Benjamin D. Miller
This quantity presents a self-contained creation to a few issues in orbit equivalence idea, a department of ergodic conception. the 1st chapters concentrate on hyperfiniteness and amenability. incorporated listed below are proofs of Dye's theorem that likelihood measure-preserving, ergodic activities of the integers are orbit an identical and of the concept of Connes-Feldman-Weiss picking amenability and hyperfiniteness for non-singular equivalence kin. The presentation here's usually motivated through descriptive set thought, and Borel and commonplace analogs of assorted effects are mentioned. the ultimate bankruptcy is an in depth account of Gaboriau's fresh effects at the concept of prices for equivalence family and teams and its purposes to proving tension theorems for activities of unfastened groups.
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Extra resources for Topics in orbit equivalence
Sample text
Sn , fn , Sn+1 associated (in the obvious way) to s. Then x = fn fn−1 · · · f0 (x)Efn fn−1 · · · f0 (y) = y , so find k with gk (x ) = y . Fix α0 ⊇ s with α0 (n+ 1) = k. Then clearly α0 ∈ A α0 (as (x, y) ∈ En+1 ), so A ∩ Ns = ∅. 13 Generic Compressibility Let E be a countable Borel equivalence relation on X, and let D : E → R+ be a Borel cocycle. E is D-aperiodic if |[x]E |x is infinite, for all x ∈ X, where | · |x is defined as in Section 8. The following extends a result of Wright [Wr]. 1 (Kechris-Miller).
Now set A = {φm (Sij ) : m < r, Si ⊆ φ−1 m (A ∩ Am ), j < pi }. Clearly A ⊆ A and µ(A \ A ) ≤ µ Sij ) : m < r, i < n} {φm (Si \ j
Given an array A, we use A to denote A0 ∪ · · · ∪ An−1 , and φi,j to denote φj ◦ φ−1 : Ai → Aj . It is useful to think of φi,j as a link between Ai , i Aj . The rank of A is given by rank(A) = n. A refinement of A of rank k is a system B = B0 , . . , Bk−1 , ψ0 , . . , ψk−1 , where {Bi } is a partition of a conull subset of A0 . This gives rise to a subarray AB of A defined by AB = B i,j ,φ i,j i,j